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# If n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is GMAT Explanation, Video Solution, and More Practice!

If n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is

A. 6
B. 12
C. 24
D. 36
E. 48

We get this one a lot in GMAT tutoring sessions. Most people fail to understand what the questions is asking and then just start working without a plan. It ends up being that work itself isn't totally off but the student doesn't really know where to take it in order to narrow down the answer choices.

What also tends to confuse is that the question asks for the largest positive integer but we end up choosing the smallest possible value of n.

## Define the question

the largest positive integer that must divide n

There it is. But let's not leave it like that. Always try to do something with the questions. In this case because we're looking for something that divides evenly into n let's set up a fraction:

n/z = integer

And we want to maximize z because we're looking for the largest integer that must divide n.

The largest integer that divides any integer is itself. So really we're looking for n. Not that important to make that inference but just wanted to point it out.

## Setup

Now let's gather the information from the question and get things set up so we can make some inferences. We know that n is a positive integer and then n squared is divisible by 72. We can write out an equation with that second piece of information.

n^2/72 = integer

We're really looking to solve for n so let's go ahead and simplify this equation.

n^2 = integer*72

Take the square root of both sides.

n = √72√integer

Now let's pull out perfect squares from 72.

n = √9√4√2√integer

n = 6√2√integer

Now we can use the first piece of information that n is a positive integer. So 6√2√integer is a positive integer. Somehow the radicals have to disappear. So √2•√integer must be an integer. What's an easy way to do that? Make integer equal 2 so you have √2•√2 = 2.

n = 6*2 = 12

What's the largest positive integer that must divide n? 12

Now, you might be thinking: is 12 the only possibility for n? Or put another way, is 2 the only possible value for the integer? Good question. No. There are an infinite number of possible values for the integer and consequently for n. Any number that cancels out the radicals will work.

√2√8 = √2√2√4 = 4*6 = 24

√2√18 = √2√2√9 = 6*6 = 36

√2√32 = √2√2√16 = 8*6 = 48

Any number that has √2•perfect square will work.

So why is the integer 2 and the correct n 12? This is coming back to what tends to confuse GMAT tutoring students (looking for the largest integer that must divide but then the answer is actually the smallest possible value of n).

Because the question is a MUST. So you need the most basic building block of n. Look at all of the values we came up with for n: 12, 24, 36, 48. What's the biggest integer they have in common (least common multiple)? 12

Regardless of which multiple of n you come up with it will always be divisible by 12.

## Additional GMAT Divisibility Practice Question

Here's another divisibility question from GMAT Official Guide: If x and y are positive integers such that y is a multiple of 5 and 3x + 4y = 200, then x must be a multiple of which of the following?

Here's a tricky exponents divisibility puzzle from GMAT question of the day

And another from the GMAT official guide that's not the same but has a similar puzzle/exponents/divisibility vibe with factorials in the mix: If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?

# If x and y are positive integers such that y is a multiple of 5 and 3x + 4y = 200, then x must be a multiple of which of the following? GMAT Explanation

If x and y are positive integers such that y is a multiple of 5 and 3x + 4y = 200, then x must be a multiple of which of the following?

A) 3
B) 6
C) 7
D) 8
E) 10

## Define the Question

x must be a multiple of which of the following? What does that actually mean? We need to figure out the basic divisibility properties of x and then use that to eliminate answer choices. Do you we need to solve for x? If we could that would be great but we don't need to and probably won't be able to.

## Setup

There are two nice ways to approach this question. There's a more practical setup and a more number properties approach. Let's go practical first.

We know that Y is a multiple of 5. So why not pick some values for Y? Then we can plug those into the equation and get some possibilities for x and see what inferences we can make.

Let's start at the minimum y multiple of 5 value (considering that both x and y are positive integers), 5, and go from there, 10, 15, 20...

We probably don't more than that.

## Solve

Now let's plug those y values into the equation: 3x + 4y = 200

3x + 20 = 200

3x = 180

x = 60

3x + 40 = 200

3x = 160

x = 160/3

3x + 60 = 200

3x = 140

x = 140/3

3x + 80 = 200

3x = 120

x = 40

Out of our 4 values of y only 5 and 20 are valid. The others produce non-integer values for x. Let's look at the results for 5 and 20 and see what inferences we can about the divisibility properties of x.

We've 40 and 60. So what's common or the least common multiple? They both have a 5 and a 4. The 3 in 60 and the 8 in 40 are not common so are not a factor that x is must have. Let's look at the answer choices and see what we can eliminate.

A) 3 Not a must have because it's not a factor of 40
B) 6 Not a must have because it's not a factor of 40
C) 7 Not a must have because it's not a factor of 40 or 60
D) 8 Not a must have because it's not a factor of 60
E) 10 With this method we can't prove that this is a must have but we've eliminated everything else and it is common between 40 and 60.

## Setup 2

The other way to do this is to simplify the equation and then make some inferences.

3x + 4y = 200

3x = 200 - 4y

3x = 4(50 - y)

## Solve 2

Keep in mind that y is a multiple of 5. So again could be 5, 10, 15, 20... What happens when you put any of those numbers in there? 50 - 5 = 45. 50 - 10 = 40. 50 - 15 = 35. You always end up with a multiple of 5. That's because of this divisibility rule:

Div by a number + Div by a number = Remain Div by that number

Div by a number + Not Div by that number = Not Div by that number

Not Div by a number + Not Div by a number = Depends on the number

It's a good rule to know.

So 3x is equal to a multiple of 5 times 4. Meaning: 3x is a multiple of 5 times 4. What about x? Well, the 3 isn't a multiple of 5 or 4 so x must be a multiple of 5 and 4. E is the only option that matches. You might say: but E (10) only has a 2 not a 4! That's OK. We're not being asked: which of the following represents all of the factors of x. We're just asked which answer choice represents factors that x must have. And x must have a 4 and a 5. So it must have a 2 and a 5.

## Additional Number Property Divisibility GMAT Practice Questions

Here's another GMAT Number Properties Puzzle question that can be cracked with basic organization: The number 75 can be written as the sum of the squares of 3 different positive integers

And here's a number properties puzzle from GMAT Question of the Day that also benefits from just getting the info from the question in shape.

# During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour and the rest of the distance at an average speed of 60 miles per hour. In terms of x, what was Francine's average speed for the entire trip? GMAT Explanation, Video Solution, and More Practice!

During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour and the rest of the distance at an average speed of 60 miles per hour. In terms of x, what was Francine's average speed for the entire trip?

A. (180−x)/2

B. (x+60)/4

C. (300−x)/5

D. 600/(115−x)

E. 12,000/(x+200)

Average speed, distance, miles per hour... Yes, it's a GMAT work and rate question. In this case we have an average rate with variables in the question and in the answer choices. That's a signal for considering picking numbers. Why pick numbers? To make life easier. How do I judge if picking numbers will make life easier? If the question would be a simple 1-2-3 if you had the actual numbers that's a pretty good indicator. Here, if you just had to calculate the average speed, assuming you know how to do a weighted average, things would be smooth. Here's another GMAT word problem from the GMAT Official Guide on which you can pick numbers to make things much easier.

Let's get back to Francine and her trip! Let's define the question: what was Francine's average speed for the entire trip?

Now let's take inventory. What numbers are we missing? Distance! Yes, I'd go ahead and pick a distance for the trip. Pick something that works well with the speeds, 40 and 60. Why not go for 240 total distance and divide that into 120 and 120 for each leg of the trip? That way you've got easy division.

120 miles/40mph = 3 hours

120 miles/60mph = 2 hours

Total time = 5 hours

Total distance = 240 miles

240/5 = 48mph

Just as a sanity check for the 48. We spent more time (3 hours) traveling at the slower speed (40mph) so it makes sense that the average speed would be closer to 40mph than 60mph.

OK, so, with total distance at 240 the average speed is 48. Of course, that's not an answer. We have to plug in "x", the total distance at an average speed of 40 miles per hour, in order to yield 48. We can use our divisibility shortcut here. Since we know the correct answer is 48 we know that the numerator has to be a multiple of 3 (because 48 is a multiple of 3). So just focus on the numerator and eliminate answer choices not divisible by 3.

A. (180−x)/2.  180-50 = 130

B. (x+60)/4.  50 + 60 = 110

C. (300−x)/5.  300-50 = 250

D. 600/(115−x) Div by 3

E. 12,000/(x+200) Div by 3

You end up with D and E as possibilities. At this point you can just work out the entire expression for both choices. You could also:

1. Continue using divisibility. 48 has four 2's in it. 600 only has 3. So that's out.
2. Use Magnitude. It should be pretty clear that D. is way too small.

So, in terms of x, what was Francine's average speed for the entire trip? E. 12,000/(x+200)

I think picking numbers is a great way to go on this one. You can also do the algebra which could be a little faster depending on how speedy you are at setting it up but, at least in my mind, is less straight forward and I've seen many GMAT tutoring students screw it up. I'll do the algebra in the diagram and in the video for your reference.

Video Solution: During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour and the rest of the distance at an average speed of 60 miles per hour. In terms of x, what was Francine's average speed for the entire trip?

## Additional GMAT Work and Rate/Weighted Average Practice

Here's a work and rate question from GMAT Question of the day which has the same basic structure and explains the rate t's a bit more.

Here's a very challenging average rate question from the GMAT Prep Tests. Same basic premise but on this one the follow through is tougher because it involves a quadratic.

GMAT Work and Rate question from the Official Guide to practice picking numbers. It's a cooperative rate so the follow through is a bit different than the above "Francine" average rate but you can still use rate T's and again pick numbers: Working simultaneously at their respective constant rates, Machines A and B produce 800 nails in x hours

# Last Sunday a certain store sold copies of Newspaper A for \$1.00 each and copies of Newspaper B for \$1.25 each, and the store sold no other newspapers that day. If r percent of the store’s revenues from newspaper sales was from Newspaper A and if p percent of the newspapers that the store sold were copies of newspaper A, which of the following expresses r in terms of p? GMAT Explanation, Video Solution, and additional practice questions!

Last Sunday a certain store sold copies of Newspaper A for \$1.00 each and copies of Newspaper B for \$1.25 each, and the store sold no other newspapers that day. If r percent of the store’s revenues from newspaper sales was from Newspaper A and if p percent of the newspapers that the store sold were copies of newspaper A, which of the following expresses r in terms of p?

A. 100p/(125–p)

B. 150p/(250–p)

C. 300p/(375–p)

D. 400p/(500–p)

E. 500p/(625–p)

OK, so this Newspaper Store question is a big one. We see this all the time in the course of GMAT tutoring. It's dense. No matter how you slice it's a bit time consuming. Understand it. Yes you should. Solve it on your GMAT? That's a maybe.

That does't mean run and hide whenever you see a dense word problem. Still, dense + variables in answer choices might be an indicator that you could be in for the long haul even if you ultimately succeed. Again: long word problem does not equal auto skip. Here's a wordy GMAT remainder/ divisibility question from the GMAT official guide that while considered tough can be solved pretty easily provided that you have a decent method.

Back to last Sunday at the newspaper store! First step: read carefully. Don't skim. Digest as you read BUT try not to solve as you read. We need to get the setup before we follow through.

Now let's go ahead and define the question. Again, let's not calculate or worry about the numbers but focus on the story (the nouns).

Here's the question: which of the following expresses r in terms of p?

Not necessarily a straightforward thing to define in a useful way. In this case I'd start by defining r and p separately and then see how we can connect them. You can start with either r or p but generally aim to start easy. I'd say p is slightly simpler so let's start there.

p = the percent of the newspapers that the store sold were copies of newspaper A

Great. Easy. So let's add some variables. #A and #B. So P = (#A/(#A + #B))*100. Why are we multiplying by 100? It's a percent! Without the 100, the fraction is just a decimal. To get the percent you need to multiply by 100. This step can range from unimportant to critical. If picking numbers and plugging back into answer choices multiplying by 100 for percents is very important. If you don't multiply by 100 your result won't match any of the answer choices.

r = the percent of the store’s revenues from newspaper sales was from Newspaper A. OK. Not bad. We don't need any additional variables for this because we have the per unit prices for Newspapers A and B.

r = (#A(1)/(#A(1) + #B(5/4)))*100

Now you could try to equate those and isolate r. r in terms of p just means, how do you transform p to equal r? That said, I'd consider picking numbers. Why? Well, if you have the numbers. Meaning, if you knew the values of r and p would it be easy to equate one with the other? Meaning, if r were 4 and p 2 can you relate those? Sure you could! r = 2p. If the question would be easier if you had the numbers then go ahead and pick some! Clearly you can't always pick numbers. There are generally two scenarios that make this possible:

1. You have variables in the answer choices
2. You have proportions in the answer choices

That doesn't mean if you are in scenario 1 or 2 that you should always pick numbers. It's just that those scenarios tend to make it possible and certainly suggest that you consider picking values.

Ok, so what variables do we have? #A and #B. We're just missing the number sold. Do we have constraints? Not really. Don't pick negative numbers or 0 because at least 1 newspaper was sold. You could pick any positive integer.

I'd stick to smaller numbers and potentially set up your expression so you create easy arithmetic. In this case we the 5/4 from the price of Newspaper B. I'd try to cancel the 4 in the denominator. Why not make #B = 4. And, while we're trying to make the GMAT easy, let's make #A also equal to 4 so we know p, the percent of the newspapers that the store sold were copies of newspaper A, is 50%. No calculation needed.

Now let's do r: (4/(4 + (5/4)*4))*100 = 400/9

Perfect. So r = 400/9. That's what we want to transform p into. Simpler way to put it: plug p (50) into the answer choices and the correct answer will yield 400/9. Now, you could do the arithmetic for all of the answer choices but there is a shortcut if you've picked numbers and are plugging back into the answers. Use divisibility. We know that the correct answer will simplify to 400/9. So I'd look at the denominator, 9. We know that the denominator if the correct answer, even before being simplified, must be a multiple of 9. So let's start by looking only at the denominators and eliminate any choice that doesn't have a denominator that is a multiple of 9.

A. 100p/(125–p). 125-50 = 75

B. 150p/(250–p). 250-50 = 200

C. 300p/(375–p) 375 - 50 = 325

D. 400p/(500–p) 500-50 = 450

E. 500p/(625–p) 625-50 = 575

Just because the denominator is divisible by 9 doesn't mean it is the correct answer. It could be that after using the divisibility trick that you're left with a couple of choices. That's fine. You can then plug in the rest for the ones you've got left. You can also use magnitude to decide between them. Sometimes choices that pass the divisibility test are either way too big or small to be the correct answer.

In terms of the algebra answer choice D means that r = 400p/(500–p).

## Additional GMAT Word Problem, Percent Change, Picking Numbers, Practice Questions

Here's a very similar tough word problem with a bunch of variables from the GMAT Official Guide: Last year the price per share of Stock X increased by

And another question from the Official Guide that's similar-ish in that you can pick numbers and use the divisibility shortcut for eliminating answer choices. It's also one that tutoring students tend to mis-interpret. During a trip, Francine traveled x percent of the total distance

Here's another very challenging word problem from the GMAT Official Guide: Last year, a certain company began manufacturing product X and sold every unit of product X that it produced. It's a data sufficiency question and neither features percent change nor picking numbers but it has the same level of density and difficulty. It requires a great setup. It's also one that you might skip (even though you should understand 100% how to get it through it in practice).

# How many of the integers that satisfy the inequality (x+2)(x+3)/(x−2) ≥ 0 are less than 5? GMAT Explanation

How many of the integers that satisfy the inequality (x+2)(x+3)/(x−2) ≥ 0 are less than 5?

A. 1
B. 2
C. 3
D. 4
E. 5

If you start doing the algebra, working the inequality/quadratic, this can get ugly. On GMAT quant remember to stay practical. Not everything has a tidy algebraic solution. Some things do so let's not completely forget about solving equations/inequalities but, again, let's just make good decisions and use the tools that make sense for the job. In this case we have a single variable inequality and a somewhat limiting constraint for the value of x: How many of the integers...less than 5.

My gut would be to start testing numbers less than 5: 4, 3, 2, 1, 0, -1, -2, -3...

The inequality we're attempting to satisfy, (x+2)(x+3)/(x−2) ≥ 0, hinges on the expression being positive or negative. With that in mind I'd pay special attention to signs. We don't really care about the actual value of the expression just whether we are above or below zero. Context is key!

Here's a video if you need to brush up on multiplying positive and negative numbers. If though you are missing that fundamental you might want to take a step back in your GMAT prep and dig back into the basics. This is a pretty advanced question to tackle if you're having trouble with signs.

OK - back to work! Popping in anything 3 or greater is positive so 4 and 3 are good. 2 yields a zero in the denominator so that's not going to work because diving by zero is undefined. -1 yields negative. -2 and -3  each yield zero so both work. -4 yields negative. So does -5 and everything smaller than that. So -2, -3, 3, and 4 all satisfy the inequality. D.

## Additional Algebra, Inequality, Positive/Negative, Picking Numbers GMAT Practice Questions

This GMAT Question of the DS Number Properties/Signs example is a bit different in the specifics especially because it's Data Sufficiency but the focus on positive/negative is spot on.

Here's the another Data Sufficiency example question with testing signs. Again, a little different but very similar in certain important aspects.

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